Inference from Inadequate and Inaccurate Data, I

Having measured D numerical properties of a physical object E which requires many more than D parameters for its complete description, we want to estimate P other numerical properties of E. Continuing the discussion in papers I(1) and II,(2) the present paper gives estimates when we believe it likely that we can guess an upper bound M on the Hilbert norm not of h(E), the model representing E in some Hilbert space, but of the orthogonal projection of h(E) onto a sufficiently large subspace. In addition, the present paper simplifies the notation of I, and makes explicit the application of Bayesian subjective probability when there are errors in the data and we want to find the joint probability distribution of more than one prediction.